How to get cursor snap status in python?
Closest Point on Mesh

Object.closest_point_on_mesh(...)

Takes a point in object space and returns the hit result, the location of hit, the normal of face hit and the face's index.

Alternatively could utilize mathutils.geometry.distance_point_to_plane(...), where each plane is defined by face center and face normal. Edit, this is somewhat flawed since the cursor can be on a faces plane but outside its edges, which would need to be tested for

Both these methods require the point to be in object space.  The 3d cursor location is in global space, convert using 

pt = o.matrix_world.inverted() @ scene.cursor.location


Here is a verbose script using both methods.  The on_face boolean is true if cursor is on face of an object.

Note testing shows the tolerance TOL (distance from point to face) needs to be in the vicinity of 5e-5

import bpy
from mathutils.geometry import distance_point_to_plane as dp2p
context = bpy.context
scene = context.scene

dg = context.evaluated_depsgraph_get()
TOL = 5e-5 # Tolerance

mesh_obs = [o for o in context.scene.objects if o.type == 'MESH']

for o in mesh_obs:
    me = o.data
    print(o.name, me.name)
    #cursor location in o's object space
    pt = o.matrix_world.inverted() @ scene.cursor.location

    # using closest point on mesh
    hit, loc, n, i = o.closest_point_on_mesh(pt, depsgraph=dg)

    print(i, (pt - loc).length)

    on_face = hit and (pt - loc).length &lt; TOL
    print("closest point on mesh", on_face)

    # using distance point to plane for each face


    for p in me.polygons:
        print(p.index, dp2p(pt, p.center, p.normal))

    on_face = any(abs(dp2p(pt, p.center, p.normal)) &lt; TOL
            for p in me.polygons)
    print("plane distance", on_face)


Result of running on default cube after clicking to snap cursor to face from view.

Cube Cube
3 2.1159816795005355e-05
closest point on mesh True
0 -0.9940524101257324
1 -1.0059475898742676
2 -1.1975289583206177
3 2.104043960571289e-05
4 -0.8024709224700928
5 -2.000020980834961
plane distance True


Related Snappping the 3D cursor to the surface and find the surface&#39;s face it was snapped to